Computer Graphics / Animation

Transcription

1 Computer Graphics / Animation Artificial object represented by the number of points in space and time (for moving, animated objects). Essential point: How do you interpolate these points in space and time? Time: Interpolation from one so-called "keyframe" to the next. "Good" interpolation allows to use few points in space and time (also key-frames with longer time intervals). Very simple interpolation: - Linear interpolation. This results in polygons during interpolation in space.

2 This works well if you want to display polygon-like objects with straight edges and corners. Problem: Does not work well on smooth objects such as circles, ellipses, landscapes, people, i.e. objects with smooth surfaces, without edges and corners. For the "smoothness" of surfaces, a definition of the consistency of surface derivations is helpful. Functions depending on variables (e.g. u or v) are required to calculate continuity: edge Mapping the plane u,v (2-dimensional, hyperplane) to the surface of our 3-D object.

3 In this way we can define continuity of surfaces: C0 Continuity: Continuity of the function itself -> no surface cracks C1 Continuity: Continuity of the first derivative -> no kinks of the surface C2 Continuity: Continuity of the second derivative -> no sudden change in curvature. Related: Geometrically defined continuity: G0, G1, G2,... independent of the parameterization of u and v. Goal usually: C2 or G2 Continuity (actually means: C0, C1, C2 Continuity) Also important for temporal interpolation to avoid artificial (robotlike) jerky movements Question: How do we get this Continuity in interpolation?

4 Nyquist sampling theorem Interpolation of samples. Also specifies how this interpolation is to be carried out: With an ideal low pass filter, with a cut-off frequency equal to half the sample frequency. The samples are our control points here Nyquist ensures that above the "Nyquist" frequency there are no components -> minimum change frequency, maximally smooth. Impulse response of the ideal low pass as interpolation function: sinc function.

5 If our samples are on the integers, they are not affected by the zero crossings of the other sinc functions. Interpolation goes through the samples/control points. By filtering with the sinc function we get a convolution with the sinc function. Convolution:

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7 Sinc function is infinite polynomial, infinitely often continuously differentiable Interpolated curve is infinitely often differentiable! Better than e.g. so-called cubic splines, which can only be continuously differentiated up to the 2nd derivative. Can this approach be applied to the multidimensional case? Yes, because we can apply this interpolation separately to each dimension, each coordinate, e.g. the x and y coordinates.

8 The x-values and the y-values are each written as functions of an auxiliary variable u. The indices of P_i specify the order: x(u_i), y(u_i) This way you could apply Nyqist interpolation to computer graphics.

9 Problem: Sinc function is infinitely long, not so practical. Alternative possibilities: Convolution of our samples with the so-called" box function" as basic function

10 no continuity. Another way to obtain continuity: Our interpolation function: box function, convolved with itself

11 Note: Interpolation or Basis function is 0 for the other samples Interpolated curve passes through our samples. C0 Continuity, but no C1 Continuity of the first derivative. Next step: same trick again, the next interpolation basis function becomes the convolution of the triangle function with itself: Cubic B-Spline widely used in Computer Animation for smooth interpolation! B stands for "Basis", basis is the above h_4(x). We want to know the polynomial h4(x) more precisely. How do we get the coefficients of the B-spline? By looking at the derivatives of h_4(x), and then integrate them, to obtain our piece-wise polynomials. We use this derivation rule:

12 4. derivation of the B-spline:

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15 For further 2 intervals: Mirror-inverted, i.e. replace u by 1-u!

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17 These polynomials are then applied to the control points (in the convolution) and we actually get our C0, C1, C2 continuous interpolation in space and time!

18 Example of a spline curve with 40 samples at an interval of 0-4 (the y-axis must be divided by 100):